Starting with Euler's remarkable identity \(\zeta (1, 2)=\zeta (3)\) , MZV's generate an algebra and they satisfy a variety of relations. A challenging problem concerning MZV's is to determine all \(\mathbb {Q}\) -linear and polynomial relations. In this direction, M. Hoffman [1]} conjectured that all multiple zeta values of a given weight can be expressed as a \(\mathbb {Q}\) -linear combination of MZV's of the same weight involving 2's and 3's.
For example, in weight 5 all MZV's are linear combinations of \(\zeta (2, 3)\)  and \(\zeta (3, 2)\) , and in weight 7, all MZV's are linear combinations of \(\zeta (2, 2, 3)\) , \(\zeta (2, 3, 2)\)  and \(\zeta (3, 2, 2)\) .
